Polyelectron problem

THE POLYELECTRON PROBLEM AND RESONANCE ENERGY The secular equation for the five resonance structures will be... 

Tie hydrogen molecule is such a small problem that all of the integrals can be written out in uU. This is rarely the case in molecular orbital calculations. Nevertheless, the same irinciples are used to determine the energy of a polyelectronic molecular system. For an ([-electron system, the Hamiltonian takes the following general form ... 

The theory of chemical shifts of the nuclei of polyelectronic atoms is complicated and certainly does not yet produce results in quantitative agreement with theory. It is conceivable that a more qualitative use of these parameters might be more appropriate to the problem in hand and an example of this sort is illustrated in Fig. 8, due to Lauterbur (75), where the chemical shifts of the Si and nuclei in analogous com-... 

The electron correlation problem occurs with all polyelectronic atoms. To treat these systems using the quantum mechanical model, we must make approximations. The simplest approximation involves treating each electron as if it were moving in a field of charge that is the net result of the nuclear attraction and the average repulsions of all the other electrons. To see how this is done, let s compare the neutral helium atom and the He+ ion ... 

Nothing we do will allow us to solve the problem exactly, because the electron motions are correlated. That is, because electrons repel each other, the movement of a given electron will affect the movements of all the others. This correlation problem is reflected in the Schrodinger equation for polyelectronic atoms in the following way. Because the equation contains energy terms that simultaneously involve two different electrons, it cannot be separated rigorously into equations that involve only one electron. Thus the Schrodinger equation for polyelectronic atoms cannot be solved exactly. 

However, even though it is formulated rather easily, this problem cannot be solved exactly. The difficulty is the same as that encountered in dealing with polyelectronic atoms—the electron correlation problem. Since we cannot account for the details of the electron movements, we cannot deal with the electron-electron interactions in a specific way. We need to make approximations that allow the solution of the problem but that do not destroy the model s physical integrity. The success of these approximations can be measured only by comparing predictions from the theory with experimental observations. In this case we will see that the simplified model works well. 

Having defined the four quantum numbers n, /, m and s which describe the electron, we may now return to the problem of the possible electronic states of a polyelectron atom. These are defined by the Pauli exclusion principle, which states that no two electrons in the same atom can possess... 

The problem of the structure of the hydrogen atom is the most important problem in the field of atomic and molecular structure, not only because the theoretical treatment of this atom is simpler than that of other atoms and of molecules, but also because it forms the basis for the discussion of more complex atomic systems. The wave-mechanical treatment of polyelectronic atoms and of molecules is usually closely related in procedure to that of the hydrogen atom, often being based on the use of hydrogenlike or closely related wave functions. Moreover, almost without exception the applications of qualitative and semiquantitative wave-mechanical arguments to chemistry involve the functions which occur in the treatment of the hydrogen atom. 

At a more theoretical level, nucleophilic reactivity can be treated by the polyelectron perturbation method (25), which in its simplest form gives the perturbation energy in terms of a first-order (Coulombic) term and a second-order term involving orbital energies aand ak (Figure 2). Subsequently, this equation was adapted by Hudson and Filippini (27) and independently by Klopman et al. (28) to the problem of enhanced nucleophilic reactivity. This problem is particularly thorny, and it will now be discussed in some detail. 

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